LINEAR RECURRENCE RELATIONS FOR CLUSTER VARIABLES …bernhard.keller/publ/KellerScherotzk... · 1) Our main theorem can be viewed as generalizing a result from section 9 of [20] on

LINEAR RECURRENCE RELATIONS FOR CLUSTER

VARIABLES OF AFFINE QUIVERS

BERNHARD KELLER AND SARAH SCHEROTZKE

Abstract. We prove that the frieze sequences of cluster variables as-sociated with the vertices of an affine quiver satisfy linear recurrencerelations. In particular, we obtain a proof of a recent conjecture byAssem-Reutenauer-Smith.

1. Introduction

Caldero and Chapoton noted in [6] that one obtains natural generaliza-

tions of Coxeter-Conway’s frieze patterns [11] [9] [10] when one constructs

the bipartite belt of the Fomin-Zelevinsky cluster algebra [17] associated

with a (connected) acyclic quiver Q. Such a generalized frieze pattern con-

sists of a family of sequences of cluster variables, one sequence for each

vertex of the quiver. For simplicity, we call these sequences the frieze se-

quences associated with the vertices of Q. Recently, they have been studied

by Assem-Reutenauer-Smith [2] and by Assem-Dupont [1] for affine quivers

Q. They also appear implicitly in the work of Di Francesco and Kedem, cf.

for example [14] [13].

Our main motivation in this paper comes from a conjecture formulated by

Assem, Reutenauer and Smith [2]: They proved that if the frieze sequences

associated with a (valued) quiver Q satisfy linear recurrence relations, then

Q is necessarily affine or Dynkin. They conjectured that conversely, the

frieze sequences associated with a quiver of Dynkin or affine type always

satisfy linear recurrence relations. For Dynkin quivers, this is immediate

from Fomin-Zelevinsky’s classification theorem for the finite-type cluster al-

gebras [18]. In [2], Assem-Reutenauer-Smith gave an ingenious proof for the

affine types A and D as well as for the non simply laced types obtained from

these by folding. For the exceptional affine types, the conjecture remained

open.

In this paper, we prove Assem-Reutenauer-Smith’s conjecture in full gen-

erality using the representation-theoretic approach to cluster algebras pio-

neered in [26]. More precisely, our main tool is the categorification of acyclic

cluster algebras via cluster categories (cf. e.g. [24]) and especially the clus-

ter multiplication formula of [7]. Our method also yields a new proof for

Key words and phrases. Cluster algebra, linear recurrence, frieze pattern, quiverrepresentation.

1

2 BERNHARD KELLER AND SARAH SCHEROTZKE

A and D. It leads to linear recurrence relations which are explicit for the

frieze sequences associated with the extending vertices and which allow us

to conjecture explicit minimal linear recurrence relations for all vertices.

Notice that in addition to their intrinsic interest, linear recurrence rela-

tions have proved useful in establishing links between cluster algebras and

canonical bases [25] and in the study of BPS spectra [8]. Finally, let us point

out two links of our results to the theory of discrete integrable systems:

1) Our main theorem can be viewed as generalizing a result from section 9

of [20] on linearisations arising from ‘period 1 quivers’, cf. section 12 of [20].

Here the linear recurrence relations are obtained using first integrals.

2) As pointed out in [2], the frieze equations (1) of section 2, which define

the frieze sequences, play the exact same role as the corresponding T - and

Q-systems in [14] [13]. Those systems are integrable and this property was

used in [14] [13] to derive linear recurrence relations. Namely, the (constant)

coefficients of these relations were constructed as conserved quantities for

the system. Similarly, in our context, the crucial element Xδ (defined just

before Theorem 5.5) plays the role of a conserved quantity. We thank the

referee for raising the interesting question of whether the equations (1) of

section 2 may also be interpreted as an integrable system. We hope to come

back to this in a future article.

Acknowledgments

The first-named author thanks Andre Joyal and Christophe Reutenauer

for their kind hospitality at the Laboratoire de combinatoire et d’informa-

tique mathematique at the Universite du Quebec a Montreal, where he first

learned about the problem at the origin of this article. He is grateful to

Andy Hone for an interesting conversation on the subject of this article

and to Robert Marsh for pointing out the connection to the results of [20].

The second-named author thanks Alastair King for a helpful discussion and

La Fondation Sciences Mathematiques de Paris for their financial support.

Both authors thank the referee for pointing out the connection to discrete

integrable systems.

2. Main result and plan of the paper

Let Q be a finite quiver without oriented cycles. We assume that its

vertices are numbered from 0 to n in such a way that each vertex i is a sink

in the full subquiver on the vertices 1, . . . , i. We introduce a total order

on the set N×Q0 by requiring that (j, i) ≤ (j′, i′) if we have j < j′ or both

j = j′ and i ≥ i′ hold.

The generalized frieze pattern associated with Q is a family of sequences

(Xij)j∈N of elements of the field Q(x0, . . . , xn), where i runs through the

vertices of Q. We recursively define these frieze sequences as follows: we set

LINEAR RECURRENCE RELATIONS 3

Xi0 = xi for all vertices i of the quiver. Once Xi′

j′ has been defined for every

pair (j′, i′) < (j + 1, i), we define Xij+1 by the equality

(1) Xij+1X

ij = 1 +

∏s→i

Xsj+1

∏i→d

Xdj .

Note that the elements of the frieze sequences defined in this way are cluster

variables of the cluster algebra AQ associated with Q, cf. [6] [24]. The aim

of this paper is to show the following result:

Theorem 2.1. If Q is an affine quiver, then every frieze sequence (Xij)j∈N

satisfies a linear recurrence relation.

This confirms the main conjecture of Assem-Reutenauer-Smith’s [2]. They

proved it for the case where Q is of type A or D (and for the non simply

laced types obtained from these by folding). We will provide a new proof

for these types and an extension to the exceptional types. Following [2] we

show in section 9, using the folding technique, that the theorem also holds

for affine valued quivers.

Our proof is based on the additive categorification of the cluster algebra

AQ by the cluster category of Q as introduced in [4]. In addition to the

cluster category, the main ingredient of our proof is the Caldero-Chapoton

map [6], which takes each object of the cluster category to an element of

the field Q(x0, . . . , xn). Under this map, the exchange relations used to

define the cluster variables are related to certain pairs of triangles in the

cluster category, called exchange triangles. We will obtain linear recurrence

relations from ‘generalized exchange relations’ obtained via the Caldero-

Chapoton map from ‘generalized exchange triangles’.

The main steps of the proof are as follows:

Step 1. We describe the action of the Coxeter transformation on the root

system of an affine quiver.

Step 2. We show that the frieze sequence associated with a vertex i of

the quiver is the image under the Caldero-Chapoton map of the τ -orbit of

the projective indecomposable module associated with the vertex i.

Step 3. We prove the existence of generalized exchange triangles in the

cluster category of an affine quiver using Step 1.

Step 4. By Step 2 we can deduce relations between the frieze sequences

associated with vertices of the quiver from the generalized exchange triangles

constructed in Step 3.

Step 5. The relations between frieze sequences obtained in Step 4 are

either linear recurrence relations or they show that a fieze sequence is a

product or sum of sequences that satisfy a linear recurrence. Hence all

frieze sequences satisfy a linear recurrence relation.

In section 1, we study the action of the Coxeter transformation c of an

affine quiver on the roots corresponding to preprojective indecomposables.


We use this result to determine, for every affine quiver, the minimal strictly

positive integers b and m such that c satisfies cb = id +m〈−, δ〉δ, where

〈−,−〉 denotes the Euler form of the quiver. Let us stress that b is not the

Coxeter number of the associated finite root system.

In section 2, we briefly recall the cluster category of a quiver without

oriented cycles. We introduce the Caldero-Chapoton map from the class of

objects of the cluster category to Q(x0, . . . , xn) and define exchange trian-

gles and generalized exchange triangles of the cluster category. We state a

result which describes how a pair of exchange triangles determines an equa-

tion between the images of the objects appearing in the triangles under the

Caldero-Chapoton map. Then we show that the frieze sequence associated

with a vertex i is obtained by applying the Caldero-Chapoton map to the

τ -orbit of the projective indecomposable module associated with i viewed

as an object in the cluster category.

In the third section, we give conditions, in the case of an affine quiver,

for the existence of certain generalized exchange triangles. We deduce linear

recurrence relations from these generalised exchange triangles, using the

results of the previous section.

In the next three sections, we show that the main theorem holds for affine

quivers. In doing so we use results of section 1 to show that the conditions of

section 3 are satisfied. The exchange triangles yield relations between frieze

sequences that prove that the sequences satisfy linear recurrence relations.

In section 6, we prove the main theorem for affine quivers of type D and

in section 7 for all exceptional affine quivers. Here the linear recurrence

relations are given explicitly for frieze sequences associated with extending

vertices. For all other frieze sequences, the existence of a linear recurrence

is proven by showing that every sequence associated with a vertex can be

written as a product or a linear combination of sequences satisfying a linear

recurrence.

In section 8, we prove the main theorem for affine quivers of type Ap,q.

Here the explicit linear recurrence relations are given only if p equals q.

Otherwise, the existence of a linear recurrence relation is shown simulta-

neously for all frieze sequences by considering the sequence of vectors in

Q(x0, . . . , xn)n+1 whose ith coordinate is given by the entries of the frieze

sequence associated with the vertex i for all vertices i of the quiver.

In section 9, we extend the main theorem to valued quivers using the

folding technique and in the final section 10, we conjecture explicit minimal

linear recurrence relations.

3. On the Coxeter transformation of an affine quiver

We first fix the notation and recall some basic facts. We refer to [12] and

[3] for an introduction to quivers and their representations.


Let Q be an affine quiver, i.e. a quiver whose underlying graph is an

extended simply laced Dynkin diagram ∆. The type of Q is the diagram

∆ except if we have ∆ = An, in which case the type of Q is Ap,q where,

for a chosen cyclic orientation of the underlying graph of Q, the number of

positively (respectively negatively) oriented arrows equals p (respectively q).

We number the vertices of Q from 0 to n and define the Euler form of Q as

the bilinear form 〈, 〉 on Zn+1 such that, for a and b in Zn+1, we have

〈a, b〉 =n∑i=0

aibi −n∑

i,j=0

cjiaibj ,

where cij is the number of arrows from i to j in Q. The symmetrized Euler

form (, ) is defined by

(a, b) = 〈a, b〉+ 〈b, a〉

for a and b in Zn+1. A root for Q is a non zero vector α in Zn+1 such that

(α, α)/2 ≤ 1; it is real if we have (α, α)/2 = 1 and imaginary if (α, α) = 0.

It is positive if all of its components are positive. The root system Φ is the

set of all roots. There is a unique root δ with strictly positive coefficients

whose integer multiples form the radical of the form (, ) (cf. Chapter 4 of

[12]). A vertex i of Q is an extending vertex if we have δi = 1. If α is a real

root, the reflection at α is the automorphism sα of Zn+1 defined by

sα(x) = x− (α, x)α.

For each vertex i, the simple root αi is the (i+ 1)th vector of the standard

basis of Zn+1. Let us number the vertices in such a way that each vertex

i is a sink of the full subquiver of Q on the vertices 0, . . . , i. Using this

ordering, we define the Coxeter transformation of Q to be the composition

c = sα0sα1 · · · sαn .

We have

〈x, y〉 = −〈y, cx〉

for all x and y in Zn+1.

Let k be an algebraically closed field and kQ the path algebra of Q over

k. Let mod kQ be the category of k-finite-dimensional right kQ-modules.

For a vertex i of Q, we denote the simple module supported at i by Si, its

projective cover by Pi and its injective hull by Ii. The map taking a module

M to its dimension vector

dimM = (dim Hom(Pi,M))i=0...n

induces an isomorphism from the Grothendieck group of mod kQ to Zn+1.

By Kac’s theorem, the dimension vectors of the indecomposable modules

are precisely the positive roots. For two modules L and M , we have

〈dimL,dimM〉 = dim Hom(L,M)− dim Ext1(L,M).


For an indecomposable non injective module M , we have

c−1 dimM = dim τ−1m M ,

where τ is the Auslander-Reiten translation of the module category mod kQ.

Theorem 3.1. There exist a strictly positive integer b and a non zero integer

m such that cb = id−m〈−, δ〉δ. The integer b is a multiple of the width of

the tubes in the Auslander-Reiten quiver of Q.

(1) For Q of type Et, the minimal b is given by b = 6 for t = 6; b = 12

for t = 7 and b = 30 for t = 8. In all those cases m is equal to 1.

(2) For Q of type Dn, we have for even n that b = n− 2 and m = 1; if

n is odd, we have b = 2n− 4 and m = 2.

(3) For Q of type Ap,q, the minimal b is the least common multiple of

p and q and m is the order of the class of q in the additive group

Z /(p+ q)Z.

We will give a uniform interpretation of the integer b in Lemma 3.2 below.

Let us stress that, contrary to a common misconception, it is not the Coxeter

number of the corresponding finite root system.

Proof. The automorphism induced by c permutes the elements of the image

of Φ ∪ {0} in Zn+1 /Z δ. This image is finite (see [12, 7]) and generates

Zn+1 /Z δ. Therefore there exists a strictly positive integer b such that cb

induces the identity on Zn+1 /Z δ. It follows that there is a linear form

f : Zn+1 → Z such that cb − id is equal to 〈f,−〉δ. In order to show that

f is a multiple of 〈−, δ〉, as 〈−, δ〉 is primitive, it is sufficient to show that

f vanishes on the kernel of 〈−, δ〉. By [12, 7] the kernel is generated by the

dimension vectors of the regular modules. Clearly it is enough to verify that

f vanishes on the dimension vectors of the regular simple modules. Let M

be such a module. If M lies in a homogenous tube, its dimension vector

is δ and f(δ) vanishes by construction. Let us therefore assume that M is

in an exceptional tube of width s > 1. The dimension vectors of dimM ,

dim τM, . . . , dim τ s−1M are non-zero and have sum δ. It follows that they

are real roots and two by two distinct. Moreover the difference between two

of these vectors is not a non-zero multiple of δ. Therefore their images in

Zn+1 /Z δ are pairwise distinct. We must therefore have cb(dimM) = dimM

and f(dimM) vanishes. This argument also shows that the widths of the

tubes divide b.

(1) The values of b and m for the exceptional quivers can be verified by

direct computation using for example the cluster mutation applet [22].

For the other cases, we need a more detailed description of the roots and

of the Coxeter transformation. Let Q′ be the Dynkin quiver obtained from

Q by deleting the extending vertex 0 and all arrows adjacent to it. Let

α1, . . . , αn be the root basis of Q′ consisting of the dimension vectors of the


simples and let θ be the highest root of Q′. Via the inclusion of Q′ into Q

we identify the roots of Q′ with their image in Zn+1. Then the dimension

vector of the simple at the vertex 0 is α0 = δ − θ.(2) We choose the following labeling and orientation on Dn:

n

$$IIIIIIIIII 1

n− 2 // · · · // 2

??��

��>>>

>>>>

>

n− 1

::uuuuuuuuu0.

Let e1, . . . , en be the vectors in Rn+1 defined by

αi = ei − ei+1 for 1 ≤ i ≤ n− 1 and αn = en + en−1.

Then if we extend the form (−,−) to Rn+1, we have (ei, ej) = δi,j and

(ei, δ) = 0. Furthermore θ equals e1 + e2 and α0 equals δ − e1 − e2. The

reflections sαi for 1 ≤ i ≤ n− 1 act as the transposition of ei and ei+1. The

reflection at αn maps en to −en−1 and en−1 to −en. The reflection at α0 is

given by e1 7→ −e2 + δ and e2 7→ −e1 + δ.

We see that c = sα0 · · · sαn acts up to multiples of δ as the (n − 2)-cycle

on e2, . . . , en−1 and inverses the sign of e1 and en. So c maps ei to ei+1 for

2 ≤ i < n − 1 and en−1 to e2 − δ and e1 to −e1 + δ and en to −en. Then

cn−2 corresponds to the action

e1 7→{e1 if n is even−e1 + δ else.

and ei 7→ ei − δ for 2 ≤ i ≤ n− 1 and en 7→ (−1)n−2en. Therefore we have

that b equals n− 2 if n is even and b equals 2n− 4 if n is odd. We see that

cn−2 maps αn to αn − δ if n is even and it maps αn to αn−1 − δ if n is odd.

As 〈αn, δ〉 = 1, this shows that m is equal to 1 for n even and m is equal to

2 if n is odd.

(3) We consider the case Ap,q with q ≤ p. We choose the following orien-

tation and labeling on Ap,q:

0

1

44hhhhhhhhhhhhhhhhhhhhhhhhhh · · ·oo q − 1oo q //oo q + 1 // · · · // p+ q − 1

kkWWWWWWWWWWWWWWWWWWWWWWWWWW

Let E be a real vector space with basis e1, . . . , en+1, d. We endow E with a

symmetric bilinear form (−,−) such that (ei, ej) = δi,j and (d, ei) = (d, d) =

0. We have an isometric embedding of Zn+1 into E taking

αi 7→ ei − ei+1 for 1 ≤ i ≤ n and δ 7→ d.


Then θ is mapped to e1 − en+1 and α0 is mapped to d − e1 + en+1. From

now on, we identify Zn+1 with a subset of E using this embedding. The

reflection sαi acts as the transposition of ei and ei+1 for 1 ≤ i ≤ n. The

reflection at α0 maps e1 to en+1+δ and en+1 to e1−δ. Then c is given by the

product sα0 · · · sαq−1sαp+q−1 · · · sαq . The action of c is up to multiples of δ the

product of the q-cycle on e1, . . . , eq and the p-cycle on ep+q, . . . , eq+1. More

concretely, we have ei 7→ ei+1 for 1 ≤ i ≤ q − 1, eq 7→ e1 − δ and ei 7→ ei−1for q+ 2 ≤ i ≤ p+ q and eq+1 7→ ep+q + δ. Then clcm(p,q) corresponds to the

action ei 7→ ei− (lcm(p, q)/q)δ for 1 ≤ i ≤ q and ei 7→ ei + (lcm(p, q)/p)δ for

q + 1 ≤ i ≤ p+ q. Therefore we have b = lcm(p, q). We verify that clcm(p,q)

maps αq to αq− (lcm(p, q)/p+lcm(p, q)/q)δ. As 〈αq, δ〉 = 1, this shows that

m is equal to lcm(p, q)/p+ lcm(p, q)/q which is the order of the class of q in

Z /(p+ q)Z. �

We have more information when Q is of type Et for t = 6 and t = 7.

Then, for each i ∈ Q0 there are positive integers ki such that kiδi = b.

These ki satisfy cki dimPi = dimPi − δ.Let us give a uniform interpretation of the integer b of the Theorem. We

use the notations of the above proof. Let c′ denote the Coxeter transfor-

mation of the Dynkin quiver Q′. Let c be the automorphism on Zn+1 /Z δinduced by c.

Lemma 3.2. The automorphism c equals sθc′. Hence b equals the order of

the element sθc′ in the Weyl group of Q′.

Proof. The embedding of the root system of Q′ in Q given in the proof of

3.1 yields an embedding of the Weyl group of Q′ into the Weyl group of Q

such that every reflection at a root of Q′ fixes δ. Hence c equals sδ−θc′. We

can write every element y ∈ Zn+1 as a linear combination of the roots of Q′

and δ. Let y = j + tδ where j is a linear combination of the roots of Q′ and

t ∈ Z. Then

sδ−θ(j + tδ) = j − (θ, j)θ + (t+ (θ, j))δ = sθ(j) + (t+ (θ, j))δ.

Therefore the action of sδ−θ modulo δ equals the action of sθ. As c′ fixes δ,

we have c = sθc′ and b is the order of sθc

′. �

We denote by σ the automorphism on Dn with σ1 = 0, σ0 = 1 and

σn = n − 1, σ(n − 1) = n and σ fixes all other vertices of Dn. Recall that

the extending vertices of Dn are precisely 0, 1, n− 1 and n.

Lemma 3.3. (a) Let Q be of type Dn. Suppose that n is odd and i is an

extending vertex of Q. Then cn−2(dimPi) = dimPσi − δ.(b) For every vertex i of Ap,q we have cli(dimPi−q) = dimPi + δ, where

li = i− q for 0 ≤ i ≤ q and li = max{q − i,−q} for q < i.


Proof. (a) We have 〈dimPi, δ〉 = δi, which equals one as i is extending.

By the proof of 3.1, we have cn−2(dimPn) = cn−2(αn) = αn−1 − δ =

dimPn−1 − δ. If we apply cn−2 to this equation, we obtain dimPn − 2δ =

c2n−4(dimPn) = cn−2(dimPn−1)− δ and thus dimPn− δ = cn−2(dimPn−1).

Furthermore cn−2(dimP1) = cn−2(∑n

i=1 αi) = cn−2(e1 + en−1) = −e1 +

δ+en−1−δ = e2+en−1+δ−θ−δ =∑n

i=2 αi+α0 = dimP0−δ. Analogously

to the first case, we apply cn−2 to the equation and obtain dimP1−δ equals

cn−2(dimP0).

(b) We first assume that i satisfies 0 < i < q. By the proof of 3.1,

we have cq−i(dimPi) = cq−i(∑q

l=i αl) = cq−i(ei − eq+1) = eq − ep+i+1 − δ =

dimPp+i−δ. For i = q we have dimPq = eq−eq+1 = dimP0−δ and for i = 0

we have cq(dimP0) = cq(eq − eq+1 + δ) = eq − δ− ep+1− δ+ δ = dimPp− δ.Let q+1 ≤ i ≤ 2q, then cq−i(dimPi) = cq−i(eq−ei+1) = ei−q−eq+1−δ =

dimPi−q − δ.Let finally 2q ≤ i ≤ p+ q − 1 if p > q, then cq(dimPi) = cq(eq − ei+1) =

eq − ei−q+1 − δ = dimPi−q − δ, which finishes the proof. �

4. Frieze sequences of cluster variables

Let F denote the field Q(x0, . . . , xn). A sequence (aj)j∈N of elements

in F satisfies a linear recurrence if for some integer s ≥ 1, there exist

elements α0, . . . , αs−1 in F such that for all j ∈ N, one has aj+s = α0aj +

. . . αs−1aj+s−1. Equivalently, the generating series∑j∈N

ajλj

in F [[λ]] is rational and its denominator is a multiple of the polynomial

P (λ) = λs − αs−1λs−1 − . . . − α0. We say that the polynomial annihilates

the sequence.

Lemma 4.1. (a) Let (aj)j∈N and (bj)j∈N be two sequences in F that satisfy

a linear recurrence relation. Then the sequences (aj + bj)j∈N and (ajbj)j∈Nsatisfy a linear recurrence relation.

(b) Let m ≥ 1 be an integer and for each 1 ≤ i ≤ m, let (aij)j∈N be a

sequence in F . We consider the sequence of vectors (vj)j∈N defined by vj =

(a1j , . . . , amj )t for all j ∈ N. Suppose there exist m×m matrices A0, . . . , As−1

over F such that for every j ∈ N we have vj+s = A0vj + . . .+ As−1vj+s−1.

Then each sequence (aij)j∈N satisfies a linear recurrence.

Proof. We refer to [5] for complete proofs of these fundamental facts. Let

us record however, that if the two series are annihilated by polynomials P

and Q, then their sum is annihilated by PQ and their Hadamard prod-

uct (ajbj)j∈N by the characteristic polynomial of CP ⊗F CQ, where CP is

the companion matrix of P . In b), the sequences are annihilated by the

determinant of the matrix λs − λs−1As−1 − . . .− λA1 −A0. �


We refer to [24] for an introduction to the links between cluster algebras

and quiver representations which we now briefly recall. Let DQ denote

the bounded derived category of kQ-modules. It is a triangulated category

and we denote its suspension functor by Σ : DQ → DQ. As kQ has finite

global dimension, Auslander-Reiten triangles exist in DQ by [21, 1.4]. We

denote the Auslander-Reiten translation of DQ by τ . On the non projective

modules, it coincides with the Auslander-Reiten translation of mod kQ. The

cluster category [4]

CQ = DQ/(τ−1Σ)Z

is the orbit category of DQ under the action of the cyclic group generated

by τ−1Σ. One can show [23] that CQ admits a canonical structure of trian-

gulated category such that the projection functor π : DQ → CQ becomes a

triangle functor.

From now on, we assume that the field k has characteristic 0. We refer to

[7] for the definition of the Caldero-Chapoton [6] map L 7→ XL from the set

of isomorphism classes of objects L of CQ to the field F . We have XτPi = xifor all vertices i of Q and XM⊕N = XMXN for all objects M and N of CQ.

We call an object M in CQ rigid if it has no self-extensions, that is if the

space Ext1CQ(M,M) vanishes.

Theorem 4.2 ([7]). a) The map L 7→ XL induces a bijection from the

set of isomorphism classes of rigid indecomposables of the cluster

category CQ onto the set of cluster variables of the cluster algebra

AQ.

b) If L and M are indecomposables such that the space Ext1(L,M) is

one-dimensional, then we have the generalized exchange relation

(2) XLXM = XE +XE′

where E and E′ are the middle terms of ‘the’ non split triangles

L // E // M // ΣL and M // E′ // L // ΣM .

Let L and M be two indecomposable objects in the cluster category such

that Ext1CQ(M,L) is one dimensional. If both L and M are rigid, then so

are E and E′ and the sequence (2) is an exchange relation of the cluster

algebra AQ associated with Q. Therefore in this case, we call the triangles

in (4.2) exchange triangles. If L or M is not rigid, we call them generalized

exchange triangles.

Corollary 4.3. For each vertex i of Q0 and each j in N, we have Xij =

Xτ−j+1Pi.

Proof. By the definition, the initial variables x0, . . . , xn are the images under

the Caldero-Chapoton map of τP0, . . . , τPn. The Auslander-Reiten compo-

nent of DQ containing the projective indecomposable modules is isomorphic


to ZQ, where the vertex (j, i) of ZQ corresponds to the isomorphism class

of τ−j+1Pi for all vertices i of Q and j ∈ Z. To prove the statement, we

use induction on the ordered set N×Q0. The claim holds for all vertices of

Q and j = 0. Now let (j, w) be a vertex of N×Q0 such that j > 0. By

the induction hypothesis, we have Xτ−j+2Pi= Xi

j−1 for all vertices i of the

quiver and Xτ−j+1Pi= Xi

j for all i > w. We consider the Auslander-Reiten

triangle ending in τ−j+1Pw

τ−j+2Pw → (⊕s→w

τ−j+2Ps)⊕ (⊕w→d

τ−j+1Pd)→ τ−j+1Pw → Στ−j+2Pw.

The three terms of this triangle are rigid and the space of extensions of

τ−1Pw by Pw is one-dimensional. By 4.2 part b), this yields the exchange

relation

Xτ−j+2PwXτ−j+1Pw

= 1 +∏w→s

Xτ−j+2Ps

∏d→w

Xτ−j+1Pd.

By the induction hypothesis, this translates into the relationXwj−1Xτ−j+1Pw

=

1+∏w→sX

sj−1∏d→wX

dj . Therefore Xτ−j+1Pw

equals Xwj , which proves the

statement. �

5. Generalized exchange triangles in the cluster category

Let Q be an affine quiver. In this section, we construct some generalized

exchange triangles in the cluster category CQ.

Lemma 5.1. Let L and N be two indecomposable preprojective kQ-modules

of defect minus one satisfying the equation dimL = dimN + δ. Then, for

every regular simple kQ-module M of dimension vector δ, there exists an

exact sequence

0→ N → L→M → 0

and dimk Ext1kQ(M,N) = 1.

Proof. As N has defect minus one, we have

−1 = 〈δ, dimN〉 = dim Hom(M,N)− dim Ext1kQ(M,N)

= −dim Ext1kQ(M,N).

By the assumption, we have dimL = dimN+δ and therefore 1 = 〈dimL, δ〉 =

dim Hom(L,M) as Ext1kQ(L,M) vanishes. Since M is regular simple, every

submodule of M that is not equal to M is preprojective. Every submodule

of L is preprojective hence of defect at most −1. Thus, every quotient of

L is of defect ≥ 0. Since the proper submodules of M are preprojective,

every non zero map from L to M is surjective. The kernel of such a map

has defect −1 and is preprojective. Therefore the kernel is indecomposable

and its dimension vector equals dimN . Any preprojective indecomposable

module is determined by its dimension vector. Thus, the kernel of every


non zero map is isomorphic to N . This proves the existence of the exact

sequence. �

Lemma 5.2. Let N and M be two kQ-modules. Then we have a canonical

isomorphism Ext1CQ(M,N) ∼= Ext1kQ(M,N)⊕DExt1kQ(N,M).

Proof. This is Proposition 1.7 c) of [4]. �

Theorem 5.3. Let i ∈ Q0 be an extending vertex and suppose there is a

positive integer b such that Pi satisfies the equation dim τ−bPi = dimPi + δ.

Then, for every regular simple kQ-module M of dimension vector δ, there

exist generalized exchange triangles in CQPi → τ−bPi →M → ΣPi and M → τ bPi → Pi → ΣM.

Proof. The defect of Pi is 〈δ, dimPi〉 = −δi, which equals −1 since i is an

extending vertex. Therefore, the defect of τ−bPi also equals −1 and the

existence of the first triangle follows from 5.1. If we rotate the first triangle,

we obtain a triangle Σ−1M → Pi → τ−bPi → M . If we apply τ b to it and

use the fact that Σ−1M ∼= τ−1M ∼= M in CQ, we get the second triangle.

By 5.1 and 5.2 the vector space Ext1CQ(M,Pi) is one-dimensional. �

Note that no indecomposable module with dimension vector δ is rigid.

Lemma 5.4. [16, 3.14] Let N and M be two regular simple kQ-modules

whose dimension vectors equal δ. Then XM equals XN .

We set Xδ = XM for any regular simple module M with dimension vector

δ. By the previous Lemma, Xδ does not depend on the choice of M .

Theorem 5.5. Let i ∈ Q0 be an extending vertex and suppose that there is

a positive integer b such that Pi satisfies the equation dim τ−bPi = dimPi +

δ. Then the frieze sequence (Xij)j∈Z satisfies the linear recurrence relation

XδXij = Xi

j−b +Xij+b for all j ∈ Z.

Proof. Applying τ−j to the generalized exchange triangles of 5.3 gives new

generalized exchange triangles of the form

τ−jPi → τ−j−bPi →M → τ−jΣPi and M → τ b−jPi → τ−jPi → ΣM

since τM is isomorphic to M . These generalized exchange triangles yield the

linear recurrence relation XδXij = Xi

j−b +Xij+b for all j ≥ b by 4.2 b). �

6. Type D

Let Q be of type Dn. We use the same orientation and labeling of Dn as

in the proof of 3.1.

Theorem 6.1. Let n be even and let i be an extending vertex of Q. Then

the frieze sequence (Xij)j∈Z satisfies the linear recurrence relation XδX

ij =

Xij−n+2 +Xi

j+n−2 for all j ≥ n− 2.


Proof. This result follows immediately from 3.1 and 5.5. �

Theorem 6.2. Suppose that n is odd and i is an extending vertex of Q.

a) For every regular simple kQ-module M with dimension vector δ, there

exist generalized exchange triangles

Pi → τ2−nPσi →M → ΣPi and M → τn−2Pσi → Pi → ΣM.

b) The frieze sequence (Xij)j∈Z satisfies the linear recurrence relation

X2δX

ij = 2Xi

j +Xij−4+2n +Xi

j−2n+4

for all j ≥ 2n− 4.

Proof. a) Using 5.1 and 3.3 there exist triangles

Pi → τ2−nPσi →M → ΣPi

and

Pσi → τ2−nPi →M → ΣPσi.

Rotating the second triangle, we get a triangle

Σ−1M → Pσi → τ2−nPi →M.

If we apply τn−2 to it and use the fact that M ∼= Σ−1M in CQ and M is

τ -periodic of period one, we get a triangle in CQ of the form

M → τn−2Pσi → Pi →M.

By 5.2 and 5.1, these are generalized exchange triangles.

b) As in the proof of 5.5 we can apply powers of τ to the triangles of a)

and we get the triangles

τ−jPi → τ2−n−jPσi →M → τ−jΣPi

and

M → τn−2−jPσi → τ−jPi →M

for all j ∈ Z. By 5.5 these triangles are generalized exchange triangles and

we obtain the relations XδXij = Xσi

n−2+j +Xσi2−n+j and XδX

σij = Xi

n−2+j +

Xi2−n+j . Multiplying the first equation with Xδ and substituting using the

second equation gives the stated recurrence relation. �

Thus we have obtained linear recurrence relations for the frieze sequences

associated with all extending vertices of the quiver Q of type Dn. Us-

ing Auslander-Reiten triangles we will now deduce the existence of linear

recurrence relations for the frieze sequences associated with neighbours of

extending vertices. There is an Auslander-Reiten triangle

Pn → Pn−2 → τ−1Pn → ΣPn.

This gives the recurrence relation for the vertex n− 2. Similarly, using the

Auslander-Reiten triangle


τ−1P1 → P2 → P1 → Στ−1P1

we obtain the recurrence relation for the vertex 2. For the vertex n − 3

we will use the following exchange triangles

Pn−2 → Pn−3 → Sn−3 → ΣPn−2

Sn−3 → Pn ⊕ Pn−1 → Pn−2 → ΣSn−3

and for 2 < i < n− 3 we will use the exchange triangles

Pi+1 → Pi → Si → ΣPi+1

Si → Pi+2 → Pi+1 → ΣSi.

These are indeed exchange triangles since we have

−1 = 〈αi, αi+1〉 = 〈αi,n∑

t=i+1

αt〉 = 〈dimSi, dimPi+1〉

= dim Hom(Si, Pi+1)− dim Ext1kQ(Si, Pi+1) = −dimk Ext1kQ(Si, Pi+1)

for all 2 < i < n− 1. We therefore obtain the relations

Xn−3j = Xn−2

j Xτ jSn−3−Xn

j Xn−1j

for all j ∈ Z and

Xij = Xi+1

j Xτ jSi−Xi+2

j

for all j ∈ Z and 2 < i < n− 2.

Note that the Si are regular modules for 2 < i < n − 3 lying in the

exceptional tube of length n−2 (cf. the tables at the end of [15]). Therefore

the corresponding frieze sequence Xτ jSiis periodic. We now use descending

induction on the vertices: we can recover linear recursion formulas for the

frieze sequence associated to a vertex i with 3 < i < n − 1 from the linear

recursion formulas of sequences associated to vertices i′ > i and the periodic

sequences Xτ jSi.

7. The exceptional types

Let Q be of type Et for t ∈ {6, 7, 8}. We will use the following labeling

and orientation:

for E6 : 1 // 2 // 7 6oo 5oo

4

OO

3

OO

,


the vector δ is given by

1 // 2 // 3 2oo 1oo

2

OO

1

OO

;

for E7 : 1 // 2 // 3 // 8 6oo 5oo 4oo

7

OO ,


1 // 2 // 3 // 4 3oo 2oo 1oo

2

OO ;

for E8 : 1 // 2 // 3 // 4 // 5 // 9 8oo 7oo

6

OO ,


1 // 2 // 3 // 4 // 5 // 6 4oo 2oo

3

OO .

Theorem 7.1. Let i be an extending vertex of Q. Then the frieze sequence

(Xij)j∈Z satisfies the linear recurrence relation

XδXij = Xi

j−b +Xij+b

for all j ∈ Z where b is as in 3.1.

Proof. This result follows immediately by 3.1 and 5.5. �

Let l ∈ Q0 be a vertex attached to an extending vertex i. Then the pro-

jective indecomposable module associated with l appears in an Auslander-

Reiten triangle

Pi → Pl → τ−1Pi → ΣPi.

This gives us the following relation between the frieze sequence associated

with the vertex l and the sequence associated with the extending vertex

X lj = Xi

jXij+1 − 1

for all j ∈ Z. By 4.1, the sequence X lj satisfies a linear recursion relation.


Let now l ∈ Q0 be a vertex such that there is an oriented path i0 =

s, . . . , it = l from an extending vertex s ∈ Q0 to l in Q of length at least

two. Then there are exchange triangles of the form

Ps → Pl → τ−1Pit−1 → ΣPs

and

τ−1Pit−1 → τ−2Pit−2 → Ps → Στ−1Pit−1 .

This gives the following relation between the sequences associated with the

vertices appearing in the oriented path

XsjX

it−1

j−1 = X lj +X

it−2

j−2

for all j ∈ Z. As all vertices connected to an extending vertex satisfy a linear

recurrence relation by the previous case, we can assume that the sequences

Xit−1

j and Xit−2

j satisfy a linear recursion using induction on the path length.

Then the sequence (X lj)j∈Z also satisfies a linear recursion relation. In the

case E6, for every non-extending vertex l of Q, there is an extending vertex

and an oriented path from the extending vertex to l. Therefore, we obtain

linear recurrence relations for all vertices of the quiver Q of type E6.

In the case E7, only the vertex labeled 7 can not be reached by an oriented

path starting in an extending vertex. In this case, we consider the exchange

triangles

P1 → τ−1P7 → τ−4P4 → τP1

and

τ−4P4 → N → P1 → τ−3P4 ,

where N is the cokernel of any non-zero morphism τ−1P1 → τ−4P4. Then

τN is the cokernel of the map P1 → τ−3P4. It is the indecomposable regular

simple module of dimension vector 001100011 which belongs to the mouth

of the tube of width 4 (cf. the tables at the end of [15]).

For E8 we use an analogous method. The vertices 6, 7 and 8 can not be

reached by an oriented path starting in an extending vertex. Therefore we

consider the following exchange triangles

P1 → τ−2P7 → τ−7P1 → τP1

and

τ−7P1 → N → P1 → τ−6P1 ,

where N is the cokernel of any non-zero morphism τ−1P1 → τ−7P1. It is

the regular simple module of dimension vector 001111001 which belongs to

the mouth of the tube of width 5 by [15, page 49]. Hence the sequence Xτ iN

is periodic. These triangles give the relation X7j = X1

j+2X1j−5 − Xτ j−2N ,


which proves that the sequence at the vertex 7 satisfies a linear recurrence

relation. The next exchange triangles are given by

P1 → τ−1P6 → τ−3P7 → τP1

and

τ−3P7 → τ−8P1 → P1 → τ−2P1.

Here the relation is X6j = X1

j+1X7j−2 − X1

j−7. Finally, the last exchange

triangles can be taken as

P1 → τ−1P8 → τ−2P6 → τP1

and

τ−2P6 → τ−4P7 → P1 → τ−1P6.

This gives the exchange relation X8j = X1

j+1X6j−1 −X7

j−3. This proves that

all frieze sequences associated with vertices of the exceptional quivers satisfy

linear recurrence relations.

8. Type Ap,q

We choose p, q ∈ N such that q ≤ p and use the same labeling and

orientation for Q as in the proof of 3.1. We view the labels of vertices

modulo p+ q. Note that all vertices of Q are extending vertices.

Theorem 8.1. (a) For every vertex i ∈ Q0 and every regular simple module

M with dimension vector δ, there are generalized exchange triangles

Pi → τ liPi−q →M → ΣPi

and

M → τ riPi+q → Pi → ΣM ,

where li = i − q for 0 ≤ i ≤ q and li = max{q − i,−q} for q < i and

ri = −li+q.(b) We obtain relations between the frieze sequences associated to the

vertices i, i+ q and i− q of the form

XδXij = Xi+q

j+li+Xi−q

j+ri

for all i ∈ Q0 and j ≥ n.

Proof. Using 5.1 and 3.3 we obtain the existence of the first triangle. If we

replace i by i+ q in the first triangle and perform a rotation, we obtain the

triangle

M → Pi+q → τ li+qPi → ΣM.

Applying τ−li+q to this triangle gives the second triangle. Exactly as in the

proof of 6.2 we can apply powers of τ to the generalized exchange triangles

and we obtain the recurrence relations stated. �


If p equals q the relation from the previous Theorem yields

XδXij = Xi+q

j+li+Xi+q

j+ri

for all i ∈ Q0 and j ≥ n as i+ q = i− q seen modulo 2q. If we iterate once,

we obtain

(X2δ − 2)Xi

j = Xij−q +Xi

j+q,

using the fact that ri − li = q for all i ∈ Q0 and j ≥ q. Hence we can see

immediately that all frieze sequences associated to vertices of the quiver Q

of type Aq,q satisfy a linear recurrence relation. In the case p 6= q we need a

different argument. We consider the sequence of vectors (v(j))j∈N given by

v(j) = (X0j , . . . , X

nj ). Then by 8.1, there are matrices A0, . . . , An such that

v(j + n + 1) =∑n

t=0Atv(j + t) for all j ∈ N. Using 4.1 b), it is clear that

the frieze sequence associated with any vertex i satisfies a linear recurrence

relation.

9. Non simply laced types

If Q is a finite quiver without oriented cycles which is endowed with a

valuation (cf. [15]), one can define frieze sequences in a natural way. We

refer to chapter 3, equation (1) of [2] for the exact definition and to the

appendix of [2] for the list of affine Dynkin diagrams, which underlie the

affined valued quivers. As in section 7.3 of [2], we can obtain the linear

recurrence relation for a frieze sequence associated with a vertex of a valued

quiver of affine type from the linear recurrence relation of a frieze sequence

associated with the vertices of a non valued affine quiver. This can be done

using the folding technique. An introduction to the folding technique can

for example be found in section 2.4 of [19].

Theorem 9.1. The frieze sequences associated with vertices of a quiver of

the type G21, G22, F41, F42, A11 or A12 satisfy linear recurrence relations.

We obtain the linear recurrence relation for a frieze sequence associated

to a vertex of a quiver of type G22 or F42 by folding E6 using the obvious

action by Z /3Z respectively Z /2Z. The linear recurrence relations in the

case F41 are obtained by folding E7 using a natural action by Z /2Z. In

the cases G21, A11 or A12, they are obtained by folding D4 using actions of

Z /3Z, Z /4Z and Z /2Z×Z /2Z respectively.

10. On the minimal linear recurrence relations

For type D4 (with the vertices numbered as in the proof of Theorem 3.1),

one checks that the following are the polynomials of the minimal linear

recurrence relations:

vertices 0, 1, 3, 4 : λ4 −Xδλ2 + 1 , vertex 2 : (λ− 1)(λ2 −Xδλ+ 1) ,


where

Xδ =x20x

21 + 2x20x

21x2 + x20x

21x

22 + 4x0x1x2x3x4 + 2x0x1x3x4

x0x1x22x3x4

+x22x

23x

24 + 2x2x

23x

24 + x23x

24

x0x1x22x3x4.

Most of the recurrence relations one obtains from our proofs are not min-

imal. However, we conjecture that for type A, they are. In the following

tables, for each vertex i of a quiver Q of type D or E, we exhibit a polyno-

mial which we conjecture to be associated with the minimal linear recurrence

relation for the frieze sequence (Xij)j∈N. Our conjecture is based on the re-

lations we have found and on numerical evidence obtained using Maple. For

an integer d and an element c of the field F = Q(x0, . . . , xn), where n+ 1 is

the number of vertices of Q, we put

P (2d, c) = λ2d − cλd + 1.

The element Xδ of the field F is always defined as after Lemma 5.4. For

type Dn, we number the vertices as in the proof of Theorem 3.1 and for the

exceptional types as in section 7. For type Dn, where n > 4 is even, we

conjecture the following minimal polynomials. Notice that the polynomials

for D4 are not obtained by specializing n to 4 in this table.

vertex degree polynomial0, 1, n− 1, n 2n− 4 P (2n− 4, Xδ)

2, . . . , n/2− 1 2n− 4 (λn−2 − 1)P (n− 2, Xδ)P (n− 2,−Xδ)

n/2 3n/2− 3 (λn/2−1 − 1)P (n− 2, Xδ)

For type Dn, where n > 3 is odd, we conjecture the following minimal

polynomials.

vertex degree polynomial0, 1, n− 1, n 4n− 8 P (4n− 8, Xδ)

n/2 2n− 4 (λn−2 − 1)P (n− 2, Xδ)

For E6, we conjecture the following minimal polynomials.

vertex degree polynomial1, 3, 5 12 P (12, Xδ)2, 4, 6 9 (λ3 − 1)P (6, Xδ)

7 16 P (4, Xδ)P (12, Xδ)


vertex degree polynomial1, 4 24 P (24, Xδ)2, 5 36 (λ12 − 1)P (12, Xδ)P (12,−Xδ)3, 6 32 P (24, Xδ)P (8, Xδ)7 18 (λ6 − 1)P (12, Xδ)8 24 (λ6 − 1)P (12, Xδ)P (6,−Xδ)



vertex degree polynomial1 60 P (60, Xδ)

2, 7 45 (λ15 − 1)P (30, Xδ)3, 6 80 P (60, Xδ)P (20, Xδ)4, 8 75 (λ15 − 1)P (30, Xδ)P (30, X2

δ − 2)5 132 P (60, Xδ)P (12, Xδ)P (60, X3

δ − 3Xδ)9 85 (λ15 − 1)P (30, Xδ)P (30, X2

δ − 2)P (10, Xδ)

Notice that for E6 and E8, the polynomial associated with a vertex i only

depends on the coefficient δi of the root δ but that the analogous statement

for E7 does not hold since the polynomials associated with the vertices 3

and 6 are different.

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B. K. : Universite Paris Diderot - Paris 7, Institut Universitaire de France,UFR de Mathematiques, Institut de Mathematiques de Jussieu, UMR 7586 duCNRS, Case 7012, Batiment Chevaleret, 75205 Paris Cedex 13, France

S. S. : Universite Paris Diderot - Paris 7, UFR de Mathematiques, Insti-tut de Mathematiques de Jussieu, UMR 7586 du CNRS, Case 7012, BatimentChevaleret, 75205 Paris Cedex 13, France

E-mail address: [email protected], [email protected]

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LINEAR RECURRENCE RELATIONS FOR CLUSTER VARIABLES …bernhard.keller/publ/KellerScherotzk... · 1) Our main theorem can be viewed as generalizing a result from section 9 of [20] on